grammaticality - Is it proper to use "that" instead of "at which" to refer to speed?

On a physics assignment, I believe that these sentences are grammatically incorrect, but some other students disagree (especially on the second one).

What is the maximum speed that the mass can be whirled without breaking the string?

What is the maximum speed that this car can round this curve without skidding

(NOTE: 33 1/3 is the frequency that it turns -- 33 1/3 revolutions per minute)

I think the use of that is incorrect, and "at which" or "that ... at" must be used to be grammatically correct. "that ..." is used above as restrictive clauses.

They accepted that the last one is incorrect.

the frequency [that it turns __ ]

Something cannot turn the "maximum speed".

For the second part, it doesn't seem like proper use of restrictive clauses.

the maximum speed [that this car can round _? this curve _?]

-> this car can round __ this curve

-> this car can round this curve __

"the maximum speed" is not the object of the clause. The car cannot round "the maximum speed".

Using "at which" or "that ... at" is better:

the maximum speed [at __ which this car can round this curve]

the maximum speed [that this car can round this curve at __ ]

Compare this to some valid examples:

Subject: This is a sentence [that I wrote __ ].

Object: This is a sentence [that __ was written by me ].

Preposition: This is a sentence [at __ which you looked ].

Detached preposition: This is a sentence [that you looked at __ ].

Am I correct?